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Cell phones

Alignments to Content Standards: F-IF.A.2


Let $f(t)$ be the number of people, in millions, who own cell phones $t$ years after 1990. Explain the meaning of the following statements.

  1. $f(10)=100.3$
  2. $f(a) = 20$
  3. $f(20) = b$
  4. $n=f(t)$

IM Commentary

This simple task assesses whether students can interpret function notation. The four parts of the task provide a logical progression of exercises for advancing understanding of function notation and how to interpret it in terms of a given context.


  1. The number of people who own cell phones in the year 2000 is $100,\!300,\!000$.
  2. There are $20,\!000,\!000$ people who own cell phones $a$ years after 1990.
  3. There will be $b$ million people who own cell phones in the year 2010.
  4. The number $n$ is the number of people (in millions) who own cell phones $t$ years after 1990.

Tricia says:

almost 4 years

A group of us just finished a task talk about this task in which some interesting ideas came up. This task at first glance seems rather straightforward, but has lots of opportunities for additional discussion if the purpose were to launch a richer discussion rather than simply check for understanding of basic function notation. Some of the ideas were: Suppose you had a graph of the function in which the y-intercept was .... write the information in functional notation and interpret it in the context of the problem; Ask the students to make some assumptions about the function based on their understanding of the context (e.g. Is the function increasing?) including perhaps asking them to sketch a graph. Based on this type of assumption one could also discuss ideas such as what a possible value of "a" was (less than 10 since there are fewer cell phone users than for f(10)?). Overall a great task that could serve many different learning and teaching goals.

Cam says:

almost 4 years

Excellent points and suggestions -- thanks for the comment!

Peter says:

almost 4 years

I'm constructing a scoring rubric for items like this. Does the following seem reasonable? (It's for part C.) I'd appreciate comments & suggestions.

(3 pts) There will be b million people who own cell phones in the year 2010.

(-1 pt) Incorrect units for function value, such as “There will be b people who own cell phones in the year 2010.”

(-1 pt) Unsimplified variable, such as “There will be b million people who own cell phones 20 years after 1990.”

(0 pts) no substantial answer

Cam says:

almost 4 years

I personally hesitate to give feedback on rubrics without knowing things like the intent of the assessment, how it will be used, etc. That said, I think penalizing students so much for the two (-1 pt) options seems extreme -- a student answering the first question actually does have a reasonably good understanding of function notation, which is the point of the exercise. I would give that 2 out of 3. Ditto for the other (-1) pt one, if not full credit -- they haven't made a single mathematical error, or arguably, no error at all. It seems quite surprising to me that you would think either of those answers less valuable than "no substantial answer".

Jennifer Lingle says:

about 4 years

Used this simple task with my 8th graders enrolled in High School Math II. It was the single best way I have ever reviewed function notation. And they came away absolutely understanding input, output, x, f(x), independent and dependent variables and how all of those terms are connected. It was fabulous.

Bill says:

about 4 years

Thanks for letting us know!